Examples of such geology occur in the Gulf of Mexico adjacent to or under salt bodies, in offshore Angola adjacent to or under salt, or in complex overthrust formations such as found in the foothills of the Canadian Rockies or the Andes of Venezuela. When imaging in such complex formations one of the basic problems is that seismic energy may reach each image point from source or receiver by a variety of raypaths. This phenomenon is referred to as “multipathing”. Current methods proposed to deal with multipath imaging are slow and expensive, and therefore impractical for widespread application to 3-D imaging using present computer technology.
Most conventional imaging methods assume there is only one path connecting the source, image point and receiver. This is true whenever the subsurface velocity behaves in a sufficiently simple manner, for example when it is controlled mostly by compaction and varies mainly in a vertical direction but only slowly in a lateral direction.
Attempts to deal with multipath imaging fall into three classes:
(a) Kirchhoff Common Offset Depth Migration
In this approach the seismic data is assembled into common offset (source-receiver distance) bins and migrated using a single preferred ray path connecting source to image point to receiver. When there are many possible ray paths, a single path is selected, often the path having the strongest amplitude, first arrival or some other preselected criterion. This imaging method is satisfactory whenever lateral velocity variation is not strong. However, multipathing is a significant effect in subsalt imaging, and many ray paths may have comparable weight in the image. Thus it often occurs that Kirchhoff migration is unable to image subsalt reflectors satisfactorily. In addition, it is inconvenient to correct Kirchhoff images for spherical divergence since this requires the relatively expensive computation of the Beylkin determinant. The theory of how to correct Kirchhoff amplitudes for spherical divergence is explained in Schleicher et al., “3-D True-Amplitude Finite-Offset Migration”, Geophysics 58, 1112–1126 (1993). Thus, Kirchhoff amplitudes are often less useful for subsequent data interpretation.
(b) Wave Equation Migration (WEM)
WEM is a more ambitious method of imaging. In a typical approach the data recorded at all the receivers for a given shot point are back-propagated using the wave equation. Energy from the shot is forward-propagated using the wave equation, and the two wave fields are cross-correlated thus producing an image as described by Claerbout in Fundamentals of Geophysical Data Processing, McGraw-Hill (1976). One difficulty with this approach is that the answers generally do not preserve seismic amplitude and therefore the resulting image cannot be directly interpreted for amplitude. This is a significant problem for the seismic interpreter because amplitudes convey information about reflection coefficients, and hence about subsurface rock properties.
Another problem with WEM is that downward wave continuation is computationally expensive. In addition the wave fields have to be computed for all of the locations downward from the source and receiver, including locations of no interest for the final image: it is difficult in WEM to use “target-oriented imaging”. (Both the Kirchhoff method and the below-described CRAM method can operate in target-oriented mode.) Thus WEM generally requires large computer resources and still takes a long time to produce a result.
(c) Common Reflection Angle Migration (CRAM)
In this approach the data are migrated into common reflection angle bins. Because rays are uniquely determined by their takeoff angle and starting point in space, CRAM enables all rays connecting source, image point, and receiver to be added into the image. In principle CRAM combines much or most of the imaging power of WEM with the relative economy of Kirchhoff migration. Xu et al. (“Common-angle migration: A strategy for imaging complex media”, Geophysics 66, 1877–1894 (2001)) describe theoretical aspects of CRAM in some detail. The method described by the authors does preserve seismic amplitude through amplitude weighting of the migrated traces. However they do not describe a computational method that is practical for application to typically sized 3-D data sets.
Koren and Kosloff (“Common reflection angle migration”, Journal of Seismic Exploration 10, 41–57 (2001)) describe an approach to CRAM. Their method proceeds image point by image point. From each image point, rays are shot up to the surface. The migrated trace added into the image is that acquired at the source and receiver locations so found. This means that the seismic data has to be read in what amounts to random order from the disk. Such a computational approach is inefficient and is only practical for 2-D imaging. However in operational practice, and especially in complex geology, one generally requires true 3-D images.
As a consequence of the limitations of the above methods, there remains a need for a method of subsalt imaging (or imaging other complex formations) that is computationally affordable and is able to provide multipath imaging in 3-D space. The present inventive method satisfies this need.